1、CHAPTER 22 Value at Risk Practice Questions Problem 22.1. Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that the daily volatilities of both assets are 1% and that the coefficient of correlation between their returns is 0.3. What is th
2、e 5-day 99% VaR for the portfolio? The standard deviation of the daily change in the investment in each asset is $1,000. The variance of the portfolios daily change is 221 0 0 0 1 0 0 0 2 0 3 1 0 0 0 1 0 0 0 2 6 0 0 0 0 0 The standard deviation of the portfolios daily change is the square root of th
3、is or $1,612.45. The standard deviation of the 5-day change is 1 6 1 2 4 5 5 $ 3 6 0 5 5 5 Because N-1(0.01) = 2.326 1% of a normal distribution lies more than 2.326 standard deviations below the mean. The 5-day 99 percent value at risk is therefore 2.3263605.55 = $8388. Problem 22.2. Describe three
4、 ways of handling interest-rate-dependent instruments when the model building approach is used to calculate VaR. How would you handle interest-rate-dependent instruments when historical simulation is used to calculate VaR? The three alternative procedures mentioned in the chapter for handling intere
5、st rates when the model building approach is used to calculate VaR involve (a) the use of the duration model, (b) the use of cash flow mapping, and (c) the use of principal components analysis. When historical simulation is used we need to assume that the change in the zero-coupon yield curve betwee
6、n Day m and Day 1m is the same as that between Day i and Day 1i for different values of i . In the case of a LIBOR, the zero curve is usually calculated from deposit rates, Eurodollar futures quotes, and swap rates. We can assume that the percentage change in each of these between Day m and Day 1m i
7、s the same as that between Day i and Day 1i . In the case of a Treasury curve it is usually calculated from the yields on Treasury instruments. Again we can assume that the percentage change in each of these between Day m and Day 1m is the same as that between Day i and Day 1i . Problem 22.3. A fina
8、ncial institution owns a portfolio of options on the U.S. dollarsterling exchange rate. The delta of the portfolio is 56.0. The current exchange rate is 1.5000. Derive an approximate linear relationship between the change in the portfolio value and the percentage change in the exchange rate. If the
9、daily volatility of the exchange rate is 0.7%, estimate the 10-day 99% VaR. The approximate relationship between the daily change in the portfolio value, P , and the daily change in the exchange rate, S , is 56PS The percentage daily change in the exchange rate, x , equals 15S . It follows that 56 1
10、 5Px or 84Px The standard deviation of x equals the daily volatility of the exchange rate, or 0.7 percent. The standard deviation of P is therefore 84 0 007 0 588 . It follows that the 10-day 99 percent VaR for the portfolio is 0 5 8 8 2 3 3 1 0 4 3 3 Problem 22.4. Suppose you know that the gamma of
11、 the portfolio in the previous question is 16.2. How does this change your estimate of the relationship between the change in the portfolio value and the percentage change in the exchange rate? The relationship is 2215 6 1 5 1 5 1 6 22P x x or 28 4 1 8 2 2 5P x x Problem 22.5. Suppose that the daily
12、 change in the value of a portfolio is, to a good approximation, linearly dependent on two factors, calculated from a principal components analysis. The delta of a portfolio with respect to the first factor is 6 and the delta with respect to the second factor is 4. The standard deviations of the fac
13、tors are 20 and 8, respectively. What is the 5-day 90% VaR? The factors calculated from a principal components analysis are uncorrelated. The daily variance of the portfolio is 2 2 2 26 2 0 4 8 1 5 4 2 4 and the daily standard deviation is 15 424 $124 19 . Since ( 1 282) 0 9N , the 5-day 90% value a
14、t risk is 1 2 4 1 9 5 1 2 8 2 $ 3 5 6 0 1 Problem 22.6. Suppose a company has a portfolio consisting of positions in stocks and bonds Assume there are no derivatives. Explain the assumptions underlying (a) the linear model and (b) the historical simulation model for calculating VaR. The linear model
15、 assumes that the percentage daily change in each market variable has a normal probability distribution. The historical simulation model assumes that the probability distribution observed for the percentage daily changes in the market variables in the past is the probability distribution that will a
16、pply over the next day. Problem 22.7. Explain how an interest rate swap is mapped into a portfolio of zero-coupon bonds with standard maturities for the purposes of a VaR calculation. When a final exchange of principal is added in, the floating side is equivalent a zero coupon bond with a maturity d
17、ate equal to the date of the next payment. The fixed side is a coupon-bearing bond, which is equivalent to a portfolio of zero-coupon bonds. The swap can therefore be mapped into a portfolio of zero-coupon bonds with maturity dates corresponding to the payment dates. Each of the zero-coupon bonds ca
18、n then be mapped into positions in the adjacent standard-maturity zero-coupon bonds. Problem 22.8. Explain the difference between Value at Risk and Expected Shortfall. Value at risk is the loss that is expected to be exceeded (100 )X% of the time in N days for specified parameter values, X and N . E
19、xpected shortfall is the expected loss conditional that the loss is greater than the Value at Risk. Problem 22.9. Explain why the linear model can provide only approximate estimates of VaR for a portfolio containing options. The change in the value of an option is not linearly related to the change
20、in the value of the underlying variables. When the change in the values of underlying variables is normal, the change in the value of the option is non-normal. The linear model assumes that it is normal and is, therefore, only an approximation. Problem 22.10. Some time ago a company has entered into
21、 a forward contract to buy 1 million for $1.5 million. The contract now has six months to maturity. The daily volatility of a six-month zero-coupon sterling bond (when its price is translated to dollars) is 0.06% and the daily volatility of a six-month zero-coupon dollar bond is 0.05%. The correlati
22、on between returns from the two bonds is 0.8. The current exchange rate is 1.53. Calculate the standard deviation of the change in the dollar value of the forward contract in one day. What is the 10-day 99% VaR? Assume that the six-month interest rate in both sterling and dollars is 5% per annum wit
23、h continuous compounding. The contract is a long position in a sterling bond combined with a short position in a dollar bond. The value of the sterling bond is 005 05153e or $1.492 million. The value of the dollar bond is 005 0515e or $1.463 million. The variance of the change in the value of the co
24、ntract in one day is 2 2 2 21 4 9 2 0 0 0 0 6 1 4 6 3 0 0 0 0 5 2 0 8 1 4 9 2 0 0 0 0 6 1 4 6 3 0 0 0 0 5 0 000000288 The standard deviation is therefore $0.000537 million. The 10-day 99% VaR is 0 0 0 0 5 3 7 1 0 2 3 3 0 0 0 3 9 6$ million. Problem 22.11. The text calculates a VaR estimate for the e
25、xample in Table 22.9 assuming two factors. How does the estimate change if you assume (a) one factor and (b) three factors. If we assume only one factor, the model is P = 0.05f1 The standard deviation of 1f is 17.55. The standard deviation of P is therefore 0.0517.55=0.8775 and the 1-day 99 percent
26、value at risk is 0.87752.326=2.0. If we assume three factors, our exposure to the third factor is 10(0.157)+4(0.256)8(0.355)7(0.195)+20.068 = 1.75 The model is therefore P = 0.05f13.87f2+1.75f3 The variance of P is 0.05217.552+3.8724.772+1.7522.082=354.8 The standard deviation of P is 84.188.354 and
27、 the 1-day 99% value at risk is 18.842.326=$43.8. The example illustrates that the relative importance of different factors depends on the portfolio being considered. Normally the second factor is less important than the first, but in this case it is much more important. Problem 22.12. A bank has a
28、portfolio of options on an asset. The delta of the options is 30 and the gamma is 5. Explain how these numbers can be interpreted. The asset price is 20 and its volatility per day is 1%. Adapt Sample Application E in the DerivaGem Application Builder software to calculate VaR. The delta of the optio
29、ns is the rate if change of the value of the options with respect to the price of the asset. When the asset price increases by a small amount the value of the options decrease by 30 times this amount. The gamma of the options is the rate of change of their delta with respect to the price of the asse
30、t. When the asset price increases by a small amount, the delta of the portfolio decreases by five times this amount. By entering 20 for S , 1% for the volatility per day, -30 for delta, -5 for gamma, and recomputing we see that ( ) 0 10EP , 2( ) 36 03EP , and 3( ) 32 415EP . The 1-day, 99% VaR given
31、 by the software for the quadratic approximation is 14.5. This is a 99% 1-day VaR. The VaR is calculated using the formulas in footnote 9 and the results in Technical Note 10. Problem 22.13. Suppose that in Problem 22.12 the vega of the portfolio is -2 per 1% change in the annual volatility. Derive
32、a model relating the change in the portfolio value in one day to delta, gamma, and vega. Explain without doing detailed calculations how you would use the model to calculate a VaR estimate. Define as the volatility per year, as the change in in one day, and w and the proportional change in in one da
33、y. We measure in as a multiple of 1% so that the current value of is 1 252 15 87 . The delta-gamma-vega model is 23 0 5 5 ( ) 2P S S or 223 0 2 0 0 5 5 2 0 ( ) 2 1 5 8 7P x x w which simplifies to 26 0 0 1 0 0 0 ( ) 3 1 7 4P x x w The change in the portfolio value now depends on two market variables
34、. Once the daily volatility of and the correlation between and S have been estimated we can estimate moments of P and use a CornishFisher expansion. Problem 22.14. The one-day 99% VaR is calculated for the four-index example in Section 22.2 as $253,385. Look at the underlying spreadsheets on the aut
35、hors website and calculate the a) the 95% one-day VaR and b) the 97% one-day VaR. The 95% one-day VaR is the 25th worst loss. This is $156,511. The 97% one-day VaR is the 15th worst loss. This is $172,224. Problem 22.15. Use the spreadsheets on the authors web site to calculate the one-day 99% VaR,
36、using the basic methodology in Section 22.2 if the four-index portfolio considered in Section 22.2 is equally divided between the four indices. In the “Scenarios” worksheet the portfolio investments are changed to 2500 in cells L2:O2. The losses are then sorted from the largest to the smallest. The
37、fifth worst loss is $238,526. This is the one-day 99% VaR. Further Questions Problem 22.16. A company has a position in bonds worth $6 million. The modified duration of the portfolio is 5.2 years. Assume that only parallel shifts in the yield curve can take place and that the standard deviation of t
38、he daily yield change (when yield is measured in percent) is 0.09. Use the duration model to estimate the 20-day 90% VaR for the portfolio. Explain carefully the weaknesses of this approach to calculating VaR. Explain two alternatives that give more accuracy. The change in the value of the portfolio
39、 for a small change y in the yield is approximately DBy where D is the duration and B is the value of the portfolio. It follows that the standard deviation of the daily change in the value of the bond portfolio equals yDB where y is the standard deviation of the daily change in the yield. In this ca
40、se 52D , 6 000 000B , and 0 0009y so that the standard deviation of the daily change in the value of the bond portfolio is 5 2 6 0 0 0 0 0 0 0 0 0 0 9 2 8 0 8 0 The 20-day 90% VaR for the portfolio is 1 2 8 2 2 8 0 8 0 2 0 1 6 0 9 9 0 or $160,990. This approach assumes that only parallel shifts in t
41、he term structure can take place. Equivalently it assumes that all rates are perfectly correlated or that only one factor drives term structure movements. Alternative more accurate approaches described in the chapter are (a) cash flow mapping and (b) a principal components analysis. Problem 22.17. C
42、onsider a position consisting of a $300,000 investment in gold and a $500,000 investment in silver. Suppose that the daily volatilities of these two assets are 1.8% and 1.2% respectively, and that the coefficient of correlation between their returns is 0.6. What is the 10-day 97.5% VaR for the portf
43、olio? By how much does diversification reduce the VaR? The variance of the portfolio (in thousands of dollars) is 2 2 2 20 0 1 8 3 0 0 0 0 1 2 5 0 0 2 3 0 0 5 0 0 0 6 0 0 1 8 0 0 1 2 1 0 4 0 4 The standard deviation is 104 04 10 2 . Since ( 1 96) 0 025N , the 1-day 97.5% VaR is 10 2 1 96 19 99 and t
44、he 10-day 97.5% VaR is 10 19 99 63 22 . The 10-day 97.5% VaR is therefore $63,220. The 10-day 97.5% value at risk for the gold investment is 5 4 0 0 1 0 1 9 6 3 3 4 7 0 . The 10-day 97.5% value at risk for the silver investment is 6 0 0 0 1 0 1 9 6 3 7 1 8 8 . The diversification benefit is 3 3 4 7
45、0 3 7 1 8 8 6 3 2 2 0 $ 7 4 3 8 Problem 22.18. Consider a portfolio of options on a single asset. Suppose that the delta of the portfolio is 12, the value of the asset is $10, and the daily volatility of the asset is 2%. Estimate the 1-day 95% VaR for the portfolio from the delta. Suppose next that
46、the gamma of the portfolio is 26 . Derive a quadratic relationship between the change in the portfolio value and the percentage change in the underlying asset price in one day. How would you use this in a Monte Carlo simulation? An approximate relationship between the daily change in the value of th
47、e portfolio, P and the proportional daily change in the value of the asset x is 1 0 1 2 1 2 0P x x The standard deviation of x is 0.02. It follows that the standard deviation of P is 2.4. The 1-day 95% VaR is 2 4 1 65 $3 96 . The quadratic relationship is 221 0 1 2 0 5 1 0 ( 2 6 )P x x or 21 2 0 1 3
48、 0P x x This could be used in conjunction with Monte Carlo simulation. We would sample values for x and use this equation to convert the x samples to P samples. Problem 22.19. A company has a long position in a two-year bond and a three-year bond as well as a short position in a five-year bond. Each bond has a principal of $100 and pays a 5%
